Question on structure of von Neumann algebras, clarification in Conway's "A course in operator theory"

Question

I was reading the section on the structure of type I von Neumann algebras in John B. Conway's "A course in operator theory" and a few questions about certain definitions and references arose, I was just seeking some clarification :).

Let $A$ be a $C^*$-algebra, $M_n(A)$ is the algebra of $n \times n$ matrices with entries in $A$, $A^{(n)}$ is the direct sum of $A$ with itself $n$ times.

On page 285 he defines:

"50.10 Definition. If $\mathcal{A}$ is a von Neumann algebra and $n$ is some cardinal number, let $M_n(\mathcal{A}) = \{ (\mathcal{A}')^{(n)} \}'$."

1 - What does this $M_n(\mathcal{A})$ have to do with the algebra of $n \times n$ matrices defined above?

Then on the next page (page 286) he writes:

"Another way to define $M_n(\mathcal{A})$ is to define the tensor product of von Neumann algebras. In this setting $M_n(\mathcal{A})$ is precisely $\mathcal{A}\otimes \mathcal{B}(\mathcal{K})$, where $\mathcal{K}$ is a Hilbert space with dimension $n$. In fact the Hilbert space $\mathcal{H} \otimes \mathcal{K}$ can be identified with $\mathcal{H}^{(n)}$. With this identification , $\mathcal{A}' \otimes \mathbb{C} \subseteq \mathcal{B}(\mathcal{H} \otimes \mathcal{K})$ is identified with $\mathcal{A}'^{(n)}$ and $M_n(\mathcal{A})$ with $\mathcal{A} \otimes \mathcal{B}(\mathcal{K})$. The reader can see the references for an exposition of the tensor product of von Neumann algebras."

I couldn't find the reference for this so I wanted to ask:

2 - In this paragraph, is this supposed to apply only to type I algebras?

3 - What does it mean for $\mathcal{A}' \otimes \mathbb{C} \subseteq \mathcal{B}(\mathcal{H} \otimes \mathcal{K})$ to be "identified" with $\mathcal{A}'^{(n)}$ and $M_n(\mathcal{A})$ with $\mathcal{A} \otimes \mathcal{B}(\mathcal{K})$? Does that mean that $\mathcal{A}' \otimes \mathbb{C}$ is isomorphic to $\mathcal{A}'^{(n)}$ and $M_n(\mathcal{A})$ is isomorphic to $\mathcal{A} \otimes \mathcal{B}(\mathcal{K})$?

Does anyone know? Or know where I can find a reference for that last paragraph?

Answer 1

I believe the problem here is the notation. Conway uses the notation

• $H^{(n)}$ to mean the $n$-fold direct sum of $H$, and
• $A^{(n)}$ for the $n$-fold direct sum of the operator $A\in B(H)$.
• $\mathcal{A}$ for the $C^*$ or von Neumann algebra.

(I didn't have Conway's book in my office, but I looked at the preview on Google Books, and searched for "cardinal number," and I found Definition 6.3.(a). I don't know if he also uses the notation $\mathcal{A}^{(n)}$ for the $n$-fold direct sum of $\mathcal{A}$, but I can see how this could be confusing!)

In what you've written, $\mathcal{A}^{(n)}$ should be the $n$-fold amplification of $\mathcal{A}$, i.e., the image of the map $$ A\mapsto \begin{pmatrix} A & & \\ & \ddots & \\ & & A \end{pmatrix} \in B(H^{(n)}). $$ So $M_n(\mathcal{A})$ is the operators on $H^{(n)}$ commuting with the diagonal action of the algebra $\mathcal{A}'$, as expected. I hope this clears up the confusion.

I think it would be more clear to use lower case letters for operators and upper case letters for algebras, e.g., $a\in \mathcal{A}$.